Category: Integration

Question Number 26781 by shubhabrata04@gmail.com last updated on 29/Dec/17 Commented by prakash jain last updated on 29/Dec/17 $${\int }_0^1(\log (1-x)-\log x)dx$$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}−{x}\right){dx}−\int_{\mathrm{0}} ^{\mathrm{1}}…

Question Number 26759 by abdo imad last updated on 29/Dec/17 $$findthevalueof{\int }_0^{\propto }\frac{dx}{(x+1)(x+2)(x+3)}.$$ Commented by abdo imad last updated on 30/Dec/17 $${we}\:{use}\:{the}\:{changement}\:{x}+\mathrm{2}={t} \…

Question Number 26756 by abdo imad last updated on 28/Dec/17 $$provethat{\int }_0^1\frac{dx}{x+e^x}=\sum _{n=0}^{\propto }\frac{{(-1)}^n}{{(n+1)}^{n+1}}{A}_n$$$${with}\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{{n}+\mathrm{1}} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:.…

Question Number 157826 by cortano last updated on 28/Oct/21 Answered by MJS_new last updated on 29/Oct/21 $$\underset{0}{\stackrel{1}{\int }}\frac{dx}{(2+5x)\sqrt[4]{2x^3(1-x)}}=$$$$\:\:\:\:\:\left[{t}=\sqrt[{\mathrm{4}}]{\frac{\mathrm{7}{x}}{\mathrm{2}\left(\mathrm{1}−{x}\right)}}\:\rightarrow\:{dx}=\mathrm{4}\sqrt[{\mathrm{4}}]{\frac{\mathrm{2}{x}^{\mathrm{3}} \left(\mathrm{1}−{x}\right)^{\mathrm{5}} }{\mathrm{7}}}{dt}\right] \…

Question Number 26738 by goswamisubhabrata007@gmail.com last updated on 28/Dec/17 Commented by prakash jain last updated on 28/Dec/17 $$\int e^x(\log x+\frac{1}{x})dx$$$$\int e^x[f\left(x\right)+f^{\prime }\left(x\right)]dx$$$$=\int{e}^{{x}} {f}\left({x}\right){dx}+\int{e}^{{x}}…